Tiêu đề 02. Complex Numbers(Q).pdf ✅

COMPLEX NUMBERS(Q) 2:1. Algebra of Complex Numbers 1. If the set R = {(a, b): a + 5b = 42, a, b ∈ N} has m elements and ∑n=1 m (1 − i n! ) = x + iy, where i = √−1, then the value of m + x + y is (a) 8 (b) 5 (c) 4 (d) 12 ( 8 th April 1 st Shift 2024) 2. Let z be a complex number such that the real part of z−2i z+2i is zero. Then, the maximum value of |z − (6 + 8i)| is equal to (a) 12 (b) ∞ (c) 8 (d) 10 (9 9 th April 2 nd Shift 2024) 3. If z is a complex number, then the number of common roots of the equations z 1985 + z 100 + 1 = 0 and z 3 + 2z 2 + 2z + 1 = 0, is equal to (a) 2 (b) 3 (c) 1 (d) 0 (30 th Jan 2 nd Shift 2024) 4. Let a ≠ b be two non-zero real numbers. Then the number of elements in the set X = {z ∈ C : Re(az 2 + bz) = a and Re(bz 2 + az) = b} is equal to (a) 2 (b) 0 (c) 3 (d) 1 (6 6 th April 2 nd Shift 2023) 5. Let A = {θ ∈ (0,2π): 1+2isinθ 1−isin θ is purely imaginary }. Then the sum of the elements in A is (a) 2π (b) 4π (c) π (d) 3π (8th April 2 nd Shift 2023) 6. The value of ( 1+sin2π 9 +icos 2π 9 1+sin2π 9 −icos 2π 9 ) 3 is (a) − 1 2 (1 − i√3) (b) 1 2 (√3 + i) (c) − 1 2 (√3 − i)

13. Let A = {θ ∈ (− π 2 , π) : 3+2isinθ 1−2isinθ is purely imaginary }. Then the sum of the elements in A is (a) 3π/4 (b) 2π/3 (c) π (d) 5π/6 14. Let z = ( √3 2 + i 2 ) 5 + ( √3 2 − i 2 ) 5 . If R(z) and I(z) respectively denote the real and imaginary parts of z, then (a) R(z) > 0 and I(z) > 0 (b) I(z) = 0 (c) R(z) < 0 and I(z) > 0 (d) R(z) = −3 (10 th Jan nd Shift 2019) 15. Let (−2 − 1 3 i) 3 = x+iy 27 (i = √−1), where x and y are real numbers, then y − x equals (a) -91 (b) -85 (c) 85 (d) 91 (11 1 th Jan 1 st Shift 2019) 16. The least positive integer n for which ( 1+i√3 1−i√3 ) n = 1, is (a) 3 (b) 5 (c) 2 (d) 6 (Online 2018) 17. Let ω be a complex number such that 2ω + 1 = z where z = √−3. If | 1 1 1 1 −ω 2 − 1 ω 2 1 ω 2 ω 7 | = 3k, then k is equal to (a) z (b) -1 (c) 1 (d) −z(2017) 18. A value of θ for which 2+3isinθ 1−2isinθ is purely imaginary is (a) π/3 (b) π/6 (c) sin−1 ( √3 4 ) (d) sin−1 ( 1 √3 ) (2016) 19. If ω(≠ 1) is a cube root of unity, and (1 + ω) 7 = A + Bω. Then (A, B) equals (a) (1,0) (b) (−1,1) (c) (0,1) (d) (1,1) (2011)